Finite Group Databasebeta

← C3×#60140 #10678747

Subgroups — C3×#60140 #10678747

Order 768.

2722 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#131C212yesmaximal, complemented
2C3×#3449C211yesmaximal, characteristic
3C3×#2932C214yesmaximal, complemented
4C3×#2931C214yesmaximal, complemented
5C3×#2930C218yesmaximal, complemented
6C3×#2929C218yesmaximal, complemented
7C3×#2905C211yesmaximal, characteristic, complemented
8C3×#2879C212yesmaximal, complemented
9C2×C3×#405C211yesmaximal, characteristic, complemented
10#60140C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×#41C2212yes—
12C2×C3×#131—216no—
13C2×C3×#131C2214yes—
14C2×C3×#131C2214yes—
15C2×C3×#131C2214yescomplemented
16C2×C3×#131C2214yescomplemented
17C22×C3×M16C2211yescharacteristic
18C22×C3×M16C2212yes—
19C22×C3×M16C2214yescomplemented
20C23×C3×C8C2211yescharacteristic
21C23×C3×C8C2214yescomplemented
22C3×#537C414yes—
23C3×#536C414yes—
24C3×#535C414yes—
25C3×#534C414yes—
26C3×#532C2214yes—
27C3×#525C2218yes—
28C2×C3×#117C2211yescharacteristic
29C3×#436C22116yescomplemented
30C3×#435C22132yescomplemented
31C3×C8×D4C22116yescomplemented
32C3×#424C2218yescomplemented
33C3×#423C2214yescomplemented
34C3×#408C2218yes—
35C3×#408C2218yescomplemented
36C3×#407C2218yescomplemented
37C3×#405C2214yescomplemented
38C3×#405C2214yescomplemented
39C2×C3×#97C2212yescomplemented
40C2×C3×C4×C8C2212yescomplemented
41C22×#131C2×C312yescomplemented
42#3449C2×C311yescharacteristic
43#2932C2×C314yescomplemented
44#2931C2×C314yescomplemented
45#2930C2×C318yescomplemented
46#2929C2×C318yescomplemented
47#2905C2×C311yescharacteristic, complemented
48#2879C2×C312yescomplemented
49C2×#405C2×C311yescharacteristic, complemented
50C2×C3×#41—28no—
51C2×C3×#41—28no—
52C2×C3×#41C2×C414yes—
53C2×C3×#41C2×C414yes—
54C2×C3×#41C2314yes—
55C2×C3×#41C2314yes—
56C22×C3×Q8C2×C412yes—
57C22×C3×D4C2×C414yes—
58C22×C3×D4C2×C412yes—
59C23×C3×C4C2314yes—
60C23×C3×C4C2311yescharacteristic
61C3×#131—216no—
62C3×#131—216no—
63C3×#131—216no—
64C3×#131—216no—
65C2×C3×M16—28no—
66C2×C3×M16—24no—
67C2×C3×M16—216no—
68C2×C3×M16C2314yes—
69C2×C3×M16C2314yes—
70C2×C3×M16C2314yes—
71C2×C3×M16C2318yes—
72C2×C3×M16C2318yes—
73C22×C3×C8—216no—
74C22×C3×C8D418yescomplemented
75C22×C3×C8C2314yes—
76C22×C3×C8C2318yes—
77C22×C3×C8C2312yes—
78C22×C3×C8C2318yes—
79C22×C3×C8C2312yes—
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2312yes—
82C22×C3×C8C2311yescharacteristic
83C3×C4⋊Q8C2×C414yes—
84C3×#127C2×C414yes—
85C3×#124C2×C418yes—
86C3×#123C2×C4116yes—
87C3×C22⋊Q8C2×C4116yes—
88C3×#121C2×C4116yes—
89C3×#121C2×C4116yes—
90C3×#120C2×C4116yes—
91C3×C4×D4C23116yes—
92C3×#117C2×C414yes—
93C3×#117C2314yes—
94C2×C3×C4⋊C4C2×C412yes—
95C2×C3×C22⋊C4C2×C412yes—
96C2×C3×C42C2312yes—
97C3×C4⋊C8C23116yes—
98C3×C22⋊C8C23132yes—
99C3×#97C2318yes—
100C3×C4×C8C2314yes—
101C3×C4×C8C2314yes—
102C22×#41C22×C312yes—
103C2×#131—216no—
104C2×#131C22×C314yes—
105C2×#131C22×C314yes—
106C2×#131C22×C314yescomplemented
107C2×#131C22×C314yescomplemented
108C22×M16C22×C311yescharacteristic
109C22×M16C22×C312yes—
110C22×M16C22×C314yescomplemented
111C23×C8C22×C311yescharacteristic
112C23×C8C22×C314yescomplemented
113#537C3×C414yes—
114#536C3×C414yes—
115#535C3×C414yes—
116#534C3×C414yes—
117#532C22×C314yes—
118#525C22×C318yes—
119C2×#117C22×C311yescharacteristic
120#436C22×C3116yescomplemented
121#435C22×C3132yescomplemented
122C8×D4C22×C3116yescomplemented
123#424C22×C318yescomplemented
124#423C22×C314yescomplemented
125#408C22×C318yes—
126#408C22×C318yescomplemented
127#407C22×C318yescomplemented
128#405C22×C314yescomplemented
129#405C22×C314yescomplemented
130C2×#97C22×C312yescomplemented
131C2×C4×C8C22×C312yescomplemented
132C24×C3C22×C414yes—
133C24×C3C22×C411yescharacteristic
134C3×#41—216no—
135C3×#41—216no—
136C3×#41—28no—
137C3×#41—28no—
138C3×#41—28no—
139C3×#41—28no—
140C2×C3×Q8—28no—
141C2×C3×Q8C22×C414yes—
142C2×C3×Q8C22×C414yes—
143C2×C3×D4—216no—
144C2×C3×D4—28no—
145C2×C3×D4C22×C418yes—
146C2×C3×D4C22×C418yes—
147C2×C3×D4C22×C414yes—
148C2×C3×D4C22×C414yes—
149C2×C3×D4C22×C4116yes—
150C22×C3×C4—28no—
151C22×C3×C4—28no—
152C22×C3×C4C22×C418yes—
153C22×C3×C4C22×C414yes—
154C22×C3×C4C22×C414yes—
155C22×C3×C4C22×C411yescharacteristic
156C22×C3×C4C22×C412yes—
157C22×C3×C4C2×D414yes—
158C22×C3×C4#4114yes—
159C22×C3×C4C2418yes—
160C22×C3×C4C2411yescharacteristic
161C22×C3×C4C2412yes—
162C3×M16—28no—
163C3×M16—28no—
164C3×M16—28no—
165C3×M16—216no—
166C3×M16—216no—
167C2×C3×C8—28no—
168C2×C3×C8—24no—
169C2×C3×C8—216no—
170C2×C3×C8—216no—
171C2×C3×C8C2×D418yes—
172C2×C3×C8C2×D418yes—
173C2×C3×C8C2×D414yes—
174C2×C3×C8C2×D414yes—
175C2×C3×C8C2414yes—
176C2×C3×C8C2418yes—
177C2×C3×C8C2411yescharacteristic
178C2×C3×C8C2412yes—
179C2×C3×C8C2411yescharacteristic
180C3×C4⋊C4C22×C418yes—
181C3×C4⋊C4C22×C4116yes—
182C3×C22⋊C4C22×C418yes—
183C3×C22⋊C4C22×C4132yes—
184C3×C42C22×C414yes—
185C3×C42C2414yes—
186C2×#41—28no—
187C2×#41—28no—
188C2×#41C2×C3×C414yes—
189C2×#41C2×C3×C414yes—
190C2×#41C23×C314yes—
191C2×#41C23×C314yes—
192C22×Q8C2×C3×C412yes—
193C22×D4C2×C3×C414yes—
194C22×D4C2×C3×C412yes—
195C23×C4C23×C314yes—
196C23×C4C23×C311yescharacteristic
197#131—216no—
198#131—216no—
199#131—216no—
200#131—216no—
201C2×M16—28no—
202C2×M16—24no—
203C2×M16—216no—
204C2×M16C23×C314yes—
205C2×M16C23×C314yes—
206C2×M16C23×C314yes—
207C2×M16C23×C318yes—
208C2×M16C23×C318yes—
209C22×C8—216no—
210C22×C8C3×D418yescomplemented
211C22×C8C23×C314yes—
212C22×C8C23×C318yes—
213C22×C8C23×C318yes—
214C22×C8C23×C312yes—
215C22×C8C23×C312yes—
216C22×C8C23×C312yes—
217C22×C8C23×C311yescharacteristic
218C22×C8C23×C311yescharacteristic
219C4⋊Q8C2×C3×C414yes—
220#127C2×C3×C414yes—
221#124C2×C3×C418yes—
222#123C2×C3×C4116yes—
223C22⋊Q8C2×C3×C4116yes—
224#121C2×C3×C4116yes—
225#121C2×C3×C4116yes—
226#120C2×C3×C4116yes—
227C4×D4C23×C3116yes—
228#117C2×C3×C414yes—
229#117C23×C314yes—
230C2×C4⋊C4C2×C3×C412yes—
231C2×C22⋊C4C2×C3×C412yes—
232C2×C42C23×C312yes—
233C4⋊C8C23×C3116yes—
234C22⋊C8C23×C3132yes—
235#97C23×C318yes—
236C4×C8C23×C314yes—
237C4×C8C23×C314yes—
238C23×C3—28no—
239C23×C3—42no—
240C23×C3—48no—
241C23×C3C4×D414yes—
242C23×C3C23×C418yes—
243C23×C3C23×C412yes—
244C23×C3C23×C411yescharacteristic
245C3×Q8—28no—
246C3×Q8—28no—
247C3×D4—216no—
248C3×D4—216no—
249C3×D4—28no—
250C3×D4—28no—
251C3×D4—416no—
252C2×C3×C4—28no—
253C2×C3×C4—216no—
254C2×C3×C4—28no—
255C2×C3×C4—28no—
256C2×C3×C4—44no—
257C2×C3×C4C4×D418yes—
258C2×C3×C4C4×D414yes—
259C2×C3×C4C23×C418yes—
260C2×C3×C4C23×C418yes—
261C2×C3×C4C23×C411yescharacteristic
262C2×C3×C4C23×C412yes—
263C2×C3×C4C22×D414yes—
264C2×C3×C4C22×D412yes—
265C2×C3×C4C2×#4112yes—
266C2×C3×C4C2×#4114yes—
267C2×C3×C4C2511yescharacteristic
268C3×C8—28no—
269C3×C8—24no—
270C3×C8C22×D412yes—
271C3×C8C22×D414yes—
272C3×C8C22×D412yes—
273C24C22×C3×C414yes—
274C24C22×C3×C411yescharacteristic
275#41—216no—
276#41—216no—
277#41—28no—
278#41—28no—
279#41—28no—
280#41—28no—
281C2×Q8—28no—
282C2×Q8C22×C3×C414yes—
283C2×Q8C22×C3×C414yes—
284C2×D4—216no—
285C2×D4—28no—
286C2×D4C22×C3×C418yes—
287C2×D4C22×C3×C418yes—
288C2×D4C22×C3×C414yes—
289C2×D4C22×C3×C414yes—
290C2×D4C22×C3×C4116yes—
291C22×C4—28no—
292C22×C4—28no—
293C22×C4C22×C3×C418yes—
294C22×C4C22×C3×C414yes—
295C22×C4C22×C3×C414yes—
296C22×C4C22×C3×C411yescharacteristic
297C22×C4C22×C3×C412yes—
298C22×C4C2×C3×D414yes—
299C22×C4C3×#4114yes—
300C22×C4C24×C318yes—
301C22×C4C24×C311yescharacteristic
302C22×C4C24×C312yes—
303M16—28no—
304M16—28no—
305M16—28no—
306M16—216no—
307M16—216no—
308C2×C8—28no—
309C2×C8—24no—
310C2×C8—216no—
311C2×C8—216no—
312C2×C8C2×C3×D418yes—
313C2×C8C2×C3×D418yes—
314C2×C8C2×C3×D414yes—
315C2×C8C2×C3×D414yes—
316C2×C8C24×C314yes—
317C2×C8C24×C318yes—
318C2×C8C24×C311yescharacteristic
319C2×C8C24×C311yescharacteristic
320C2×C8C24×C312yes—
321C4⋊C4C22×C3×C418yes—
322C4⋊C4C22×C3×C4116yes—
323C22⋊C4C22×C3×C418yes—
324C22⋊C4C22×C3×C4132yes—
325C42C22×C3×C414yes—
326C42C24×C314yes—
327C22×C3—22no—
328C22×C3—24no—
329C22×C3—28no—
330C22×C3—216no—
331C22×C3—44no—
332C22×C3—416no—
333C22×C3C2×C4×D414yes—
334C22×C3C2×C4×D412yes—
335C22×C3C24×C411yescharacteristic
336C3×C4—28no—
337C3×C4—28no—
338C3×C4C2×C4×D414yes—
339C3×C4C2×C4×D412yes—
340C3×C4C23×D411yescharacteristic
341C3×C4C22×#4111yescharacteristic
342C23—28no—
343C23—42no—
344C23—48no—
345C23C3×C4×D414yes—
346C23C23×C3×C418yes—
347C23C23×C3×C412yes—
348C23C23×C3×C411yescharacteristic
349Q8—28no—
350Q8—28no—
351D4—216no—
352D4—216no—
353D4—28no—
354D4—28no—
355D4—416no—
356C2×C4—28no—
357C2×C4—28no—
358C2×C4—28no—
359C2×C4—216no—
360C2×C4—44no—
361C2×C4C3×C4×D418yes—
362C2×C4C3×C4×D414yes—
363C2×C4C23×C3×C418yes—
364C2×C4C23×C3×C418yes—
365C2×C4C23×C3×C411yescharacteristic
366C2×C4C23×C3×C412yes—
367C2×C4C22×C3×D414yes—
368C2×C4C22×C3×D412yes—
369C2×C4C2×C3×#4112yes—
370C2×C4C2×C3×#4114yes—
371C2×C4C25×C311yescharacteristic
372C8—28no—
373C8—24no—
374C8C22×C3×D412yes—
375C8C22×C3×D414yes—
376C8C22×C3×D412yes—
377C2×C3—24no—
378C2×C3—22no—
379C2×C3—48no—
380C2×C3C22×C4×D411yescharacteristic
381C2×C3#357312yes—
382C22—22no—
383C22—24no—
384C22—28no—
385C22—216no—
386C22—44no—
387C22—416no—
388C22C2×C3×C4×D414yes—
389C22C2×C3×C4×D412yes—
390C22C24×C3×C411yescharacteristic
391C4—28no—
392C4—28no—
393C4C2×C3×C4×D414yes—
394C4C2×C3×C4×D412yes—
395C4C23×C3×D411yescharacteristic
396C4C22×C3×#4111yescharacteristic
397C3#6014011yescharacteristic, direct product factor, Sylow
398C2—24no—
399C2—22no—
400C2—48no—
401C2C22×C3×C4×D411yescharacteristic
402C2C3×#357312yes—
subgroupG/H
centerC2×C3×C8C24
commutatorC22C24×C3×C4
Frattini, cosocleC2×C4C25×C3
socleC22×C3C24×C4
prime radicalC3×#3449C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C3×C8 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C22×C3×#131C2semidirect
3. C3×#2932C2semidirect
4. C3×#2931C2semidirect
5. C3×#2930C2semidirect
6. C3×#2929C2semidirect
7. C3×#2905C2semidirect
8. C3×#2879C2semidirect
9. C2×C3×#405C2semidirect
10. #60140C3direct (N × H)
15. C2×C3×#131C22semidirect
16. C2×C3×#131C22semidirect
19. C22×C3×M16C22semidirect
21. C23×C3×C8C22semidirect
29. C3×#436C22semidirect
30. C3×#435C22semidirect
31. C3×C8×D4C22semidirect
32. C3×#424C22semidirect
33. C3×#423C22semidirect
35. C3×#408C22semidirect
36. C3×#407C22semidirect
37. C3×#405C22semidirect
38. C3×#405C22semidirect
39. C2×C3×#97C22semidirect
40. C2×C3×C4×C8C22semidirect
41. C22×#131C2×C3semidirect
43. #2932C2×C3semidirect
44. #2931C2×C3semidirect
45. #2930C2×C3semidirect
46. #2929C2×C3semidirect
47. #2905C2×C3semidirect
48. #2879C2×C3semidirect
49. C2×#405C2×C3semidirect
74. C22×C3×C8D4semidirect
106. C2×#131C22×C3semidirect
107. C2×#131C22×C3semidirect
110. C22×M16C22×C3semidirect
112. C23×C8C22×C3semidirect
120. #436C22×C3semidirect
121. #435C22×C3semidirect
122. C8×D4C22×C3semidirect
123. #424C22×C3semidirect
124. #423C22×C3semidirect
126. #408C22×C3semidirect
127. #407C22×C3semidirect
128. #405C22×C3semidirect
129. #405C22×C3semidirect
130. C2×#97C22×C3semidirect
131. C2×C4×C8C22×C3semidirect
210. C22×C8C3×D4semidirect
397. C3#60140direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×C3×#1312
C3×#34492
C3×#29322
C3×#29312
C3×#29302
C3×#29292
C3×#29052
C3×#28792
C2×C3×#4052
#601403